On the Hecke Eigenvalues of Maass Forms

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.Comment: Version 2: typos corrected and a new section on natural density adde

Impact of Parasitic and Load Current on the Attenuation of Motor Terminal Overvoltage in SiC-Based Drives

In SiC-based adjustable speed drives, the high voltage slew rate (/) of the switching transitions results in excessive overvoltage at the motor terminals due to the reflected voltages across the drive power cables. Besides the cable length, the switching rise/fall times of the voltage pulses are a key parameter to quantify the motor overvoltage in PWM inverter-fed drives. These times are varying depending on the load current and parasitic elements of SiC MOSFETs, that is, a standard two-level converter typically results in a non-uniform overvoltage envelop at the motor terminals. This article analyses the switching mechanism of the two-level converter considering the impact of SiC parasitic elements and load current showing how they affect the motor overvoltage in cable-fed drives. The analysis is then extended to the mitigation of the motor overvoltage using quasithree-level (Q3L) modulation as a candidate filter-less approach with a T-type converter. The theoretical analysis is validated through experimental tests by using the Q3L T-type converter. The analysis and results show that the instantaneous load current value critically determines the peak motor overvoltage, while it allows either a full or partial overvoltage mitigation when the Q3L modulation is adopted